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Exactness of weighted exponential systems with a defect
Abstract
Let w : (0,1) → ℝ+ be a weight. We prove that for an arbitrary Schauder basis {ei λn t}n ∈ ℤ in L2(0,1) and an arbitrary lacunary defect set A ⊂ ℤ the system {w(t) rn(t)}n ∈ ℤ ∖ A is always complete and never minimal for any weight w satisfying a natural decay condition, which is sharp on the exponential scale. Moreover, we establish a simple combinatorial criterion for systems of the form $ \{ w(t) e^{2πi n t} \}_{n \in \mathbb{Z} \setminus A} $ to be complete and minimal in L2(0,1) for an arbitrary weight w and a finite defect set A ⊂ ℤ.
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BibTeX
@misc{arefev2026exactness,
title = {{Exactness of weighted exponential systems with a defect}},
author = {Ivan Arefev and Andrei V. Semenov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06114},
eprint = {2610.06114},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06114},
}
RIS
TY - UNPB
TI - Exactness of weighted exponential systems with a defect
AU - Ivan Arefev
AU - Andrei V. Semenov
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06114
UR - https://arxiv.org/abs/2610.06114
AB - Let w : (0,1) → ℝ_+ be a weight. We prove that for an arbitrary Schauder basis {e^i λ_n t}_n ∈ ℤ in L^2(0,1) and an arbitrary lacunary defect set A ⊂ ℤ the system {w(t) r_n(t)}_n ∈ ℤ ∖ A is always complete and never minimal for any weight w satisfying a natural decay condition, which is sharp on the exponential scale. Moreover, we establish a simple combinatorial criterion for systems of the form $ \{ w(t) e^{2πi n t} \}_{n \in \mathbb{Z} \setminus A} $ to be complete and minimal in L^2(0,1) for an arbitrary weight w and a finite defect set A ⊂ ℤ.
ER -